### Multinomial Goodness of Fit Hypothesis Test **Scenario:** You are conducting a multinomial Goodness of Fit hypothesis test for the claim that all 5 categories are equally likely to be selected. That is: \[ H_0 : p_A = p_B = p_C = p_D = p_E \] ### Table Completion: Complete the table. Report all answers correct to three decimal places. | Category | Observed Count | Expected Count | Component Value | |----------|----------------|----------------|-----------------| | A | 7 | | | | B | 21 | | | | C | 19 | | | | D | 9 | | | | E | 15 | | | ### Calculations: 1. **Expected Count:** Given that the null hypothesis states that all categories are equally likely, you need to find the expected count for each category. \[ \text{Expected Count} = \frac{\text{Total Observed Count}}{\text{Number of Categories}} \] 2. **Component Value:** For each category, calculate the component value as follows: \[ \text{Component Value} = \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}} \] 3. **Total Chi-Square Test-Statistic:** Sum all the component values to get the chi-square test statistic \(\chi^2\). \[ \chi^2 = \sum \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}} \] ### Hypothesis Test Results: - **Chi-square test-statistic (\(\chi^2\)):** \[ \chi^2 = \_\_\_\_\_ \] - **p-value:** \[ \text{p-value} = \_\_\_\_\_ \] ### Conclusion: For significance level \(\alpha = 0.005\), what would be the conclusion of this hypothesis test? - [ ] Reject the Null Hypothesis - [ ] Fail to reject the Null Hypothesis **Note:** - The chi-square distribution table or software can be used to find the p-value corresponding to the calculated \